A135805 - OEIS

%I #15 Nov 11 2016 21:49:28

%S 1,0,0,120,330,6336,61776,785928,10456875,151099520,2339361024,

%T 38655753552,678721170036,12615988058880,247449420044640,

%U 5106608041235184,110596074738524661,2507849090860975488

%N Eighth column (k=7) of triangle A134832 (circular succession numbers).

%C a(n) enumerates circular permutations of {1,2,...,n+7} with exactly seven successor pairs (i,i+1). Due to cyclicity also (n+7,1) is a successor pair.

%D Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=7.

%H G. C. Greubel, <a href="/A135805/b135805.txt">Table of n, a(n) for n = 0..443</a>

%F a(n) = binomial(n+7,7)*A000757(n), n>=0.

%F E.g.f.: (d^7/dx^7) (x^7/7!)*(1-log(1-x))/e^x.

%e a(0)=1 because from the 7!/7 = 720 circular permutations of n=7 elements only one, namely (1,2,3,4,5,6,7), has seven successors.

%t f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 7], {n, 7, 25}] (* _G. C. Greubel_, Nov 10 2016 *)

%Y Cf. A135804 (column k=6), A135806 (column k=8).

%K nonn,easy

%O 0,4

%A _Wolfdieter Lang_, Jan 21 2008